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The Maximum Number of Zeros of \(r(z) - \overline{z}\) Revisited

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Abstract

Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions \(f(z) = \frac{p(z)}{q(z)} - \overline{z}\), which depend on both \(\mathrm{deg}(p)\) and \(\mathrm{deg}(q)\). Furthermore, we prove that any function that attains one of these upper bounds is regular.

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Acknowledgements

We thank an anonymous referee for several helpful suggestions, and in particular for pointing out the technical report [7].

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Authors and Affiliations

  1. TU Berlin, Institute of Mathematics, MA 4-5, Straße des 17. Juni 136, 10623, Berlin, Germany

    Jörg Liesen & Jan Zur

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  1. Jörg Liesen
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  2. Jan Zur
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Correspondence to Jörg Liesen.

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Communicated by Filippo Bracci.

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Liesen, J., Zur, J. The Maximum Number of Zeros of \(r(z) - \overline{z}\) Revisited. Comput. Methods Funct. Theory 18, 463–472 (2018). https://doi.org/10.1007/s40315-017-0231-1

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  • DOI: https://doi.org/10.1007/s40315-017-0231-1

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